For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-n endomorphism. Show that for any n-torsion point $P \in E[n]$, we must have $\alpha(P) \in E[n]$.
I know and have just showed that $[n]$ is commutative with all endomorphisms but I'm not sure how to use that fact here. What's a good way to show $\alpha(P) \in E[n]$?